1.    Find the location of all local extrema, i.e. local maximum and local minimum for the functions  and  using

a)   the First derivative test

b)   the Second derivative test

Note: need to show work completely and explain your work including the solutions as well.

Let f and g be differentiable and continuous functions for all real numbers such thatf(x) = 3::3 +91:2 +2,and 190:) = (§)x‘* + 2×3 + 1. 1. Find the location of all local extrema, i.e. local maximum and local minimum for thefunctions f and 9 using a) the First derivative testb) the Second derivative test

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Please provide full solution with the answer. This is a calculus question.

Let f ( ac ) = act – Gap ?.( a ) Locate the critical points of the given function .( b) Use the first derivative test to identify the local maxima and minima .( C ) Use the second derivative test to identify the inflection points , local maxima , and local minimal( d ) Sketch the graph of f ( ac ) .

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5.6.15. Let f : A –> B be an arbitrary function.

(a) Prove that if f is a bijection (and hence invertible), then f^-1(f(x)) = x for all x belonging to A, and f(f^-1(x)) =

x for all x belonging to B.

(b) Conversely, show that if there is a function g : B –> A, satisfying g(f(x)) = x for all x belonging to A, and

f(g(x)) = x for all x belonging to B, then f is a bijection, and f^-1 = g.

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Please help with the attached question. The question is a fill-in-the blank proof

Let E CR and F C R. Let X = {(r,y)|:r,yE IR and y = I}. Prove that ifX C E X F, thenE x F = R2 by filing out gaps in the following argument. Proof. We are given that X C _. We need to show that _ and Let (11,5) E E X F. Then, a E_ and b E_. If a. E_, then a E_ because E C_. If I; E_Tthen b E_ because F C_. Therefore, (o,b) E 13.2 because _. We showed that _. Let ((1,5) E 1R2. Then, a E_ and b E_. Therefore, by definition of X , we have (and) E Kand (b,b) E X. Since (the) E X and _, we have _. Since (b,b) E X and we have _.Therefore, (a, b) E E X F. We showed that _. _T

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2. Let Dn be the dihedral group of symmetries of a regular n-gon, and let Cn ⊂ Dn be the subgroup of rotations. Let also H ⊂ Cn be an arbitrary subgroup. Prove that H is a normal subgroup of Dn.

Remark: H is a subgroup of Cn, so it is also a subgroup of Dn. Note that H is a normal subgroup of Cn, because Cn is Abelian. This means that ghg−1 ∈ H for any g ∈ Cn. What you need to prove is a stronger result: H is normal in Dn, meaning that ghg−1 ∈ H for any g ∈ Dn, and not only for any g ∈ Cn.

Hint: One possible approach is to show that for any element g ∈ Dn, the set gHg^-1 = {ghg^-1 | h ∈ H} is a subgroup of Cn. What is the order of this subgroup?

What do we know about subgroups of cyclic groups? Another approach is to use geometric arguments to prove that for any reflection g ∈ Dn Cn and any rotation r ∈ Cn, one has grg^-1= r^-1.

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